primuscenters

Primuscenters is a term used in geometry to denote a central point associated with a finite configuration of points in Euclidean space. In its common mathematical formalization, a primuscenter is defined with respect to a finite set S = {s1, ..., sn} of sites and a corresponding set of prime numbers {p1, ..., pn} that act as weights. The primuscenter c is the minimizer of the weighted sum of squared distances E(c) = sum_{i=1}^n p_i ||c − s_i||^2. This optimization problem is strictly convex, guaranteeing a unique minimizer.

Interpretation and formula: The primuscenter is the weighted centroid of the sites, with weights given by the primes. Specifically, the minimizer is c* = (sum p_i s_i) / (sum p_i). If all p_i are equal, c* is the ordinary centroid. By choosing different prime weightings, one can bias the center toward particular sites; primes are commonly taken as 2, 3, 5, 7, … but any nonnegative prime weights yield a valid primuscenter.

Properties and computation: The primuscenter lies in the convex hull of S and depends continuously on the positions of the sites. In symmetric configurations with equal weights, the primuscenter coincides with the center of symmetry. Computationally, it reduces to a straightforward weighted average, which is O(n) to compute.

Variations and contexts: Some discussions extend the idea to higher dimensions, other distance measures, or to multi-objective variants that combine squared distances with additional penalties. The term primuscenter is not standard across mathematical literature but appears in expository accounts as an example of how prime-based weighting influences centrality.

See also: centroid, weighted centroid, center of mass, Fermat point, Weber problem.